The problem involves analyzing the graph of a function $f(x)$ and determining the graph of a new function $g(x) = f(2x)$.

Understanding the Problem:

The function $g(x) = f(2x)$ represents a horizontal compression of the function $f(x)$. Specifically, every point on the graph of $f(x)$ is moved closer to the y-axis by a factor of 2.

Step-by-Step Explanation:

1. Horizontal Compression: The transformation $f(2x)$ compresses the graph of $f(x)$ horizontally by a factor of 2. This means that for any point $(x, y)$ on the graph of $f(x)$, there is a corresponding point $(\frac{x}{2}, y)$ on the graph of $g(x)$.

2. Applying to the Graph:

   • The vertex of the parabola $f(x)$, originally at $(4, -4)$, will now be at $\left(\frac{4}{2}, -4\right) = (2, -4)$ for $g(x)$.
   • The point $(2, 0)$ on the graph of $f(x)$ will now be at $(1, 0)$ on the graph of $g(x)$.
   • Similarly, the point $(6, 0)$ on the graph of $f(x)$ will be at $(3, 0)$ on the graph of $g(x)$.

3. Graph of $g(x)$:

   • The graph of $g(x)$ will still be a parabola, but it will be narrower than the graph of $f(x)$.
   • The x-intercepts of $g(x)$ will be at $x = 1$ and $x = 3$, and the vertex will be at $(2, -4)$.

Summary:

The graph of $g(x) = f(2x)$ will be a horizontally compressed version of the graph of $f(x)$ with the vertex at $(2, -4)$ and x-intercepts at $x = 1$ and $x = 3$.

Would you like to see a visual representation of the graph of $g(x)$? Do you have any questions or would you like further details?

Here are some related questions you might consider:

1. How would the graph change if $g(x)$ were defined as $g(x) = f\left(\frac{x}{2}\right)$?
2. What happens to the graph if $g(x) = 2f(2x)$?
3. How do vertical shifts affect the graph of a function?
4. What does the transformation $g(x) = f(x - 2)$ do to the graph of $f(x)$?
5. How can you determine the new x-intercepts after a horizontal compression?

Tip: Remember, horizontal transformations like $f(2x)$ affect the x-values of the function, compressing or stretching the graph horizontally.